Internal problem ID [8515]
Internal file name [OUTPUT/7448_Sunday_June_05_2022_10_55_20_PM_49527455/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 179.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[_rational, _Riccati]
\[ \boxed {3 x \left (x^{2}-1\right ) y^{\prime }+x y^{2}-\left (x^{2}+1\right ) y=3 x} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {-x^{2} y +x \,y^{2}-3 x -y}{3 x \left (x^{2}-1\right )} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {x y}{3 x^{2}-3}-\frac {y^{2}}{3 \left (x^{2}-1\right )}+\frac {1}{x^{2}-1}+\frac {y}{3 x \left (x^{2}-1\right )} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {1}{x^{2}-1}\), \(f_1(x)=-\frac {-x^{2}-1}{3 x \left (x^{2}-1\right )}\) and \(f_2(x)=-\frac {1}{3 \left (x^{2}-1\right )}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {u}{3 \left (x^{2}-1\right )}} \tag {1} \end {align*}
Using the above substitution in the given ODE results (after some simplification)in a second order ODE to solve for \(u(x)\) which is \begin {align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end {align*}
But \begin {align*} f_2' &=\frac {2 x}{3 \left (x^{2}-1\right )^{2}}\\ f_1 f_2 &=\frac {-x^{2}-1}{9 x \left (x^{2}-1\right )^{2}}\\ f_2^2 f_0 &=\frac {1}{9 \left (x^{2}-1\right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -\frac {u^{\prime \prime }\left (x \right )}{3 \left (x^{2}-1\right )}-\left (\frac {2 x}{3 \left (x^{2}-1\right )^{2}}+\frac {-x^{2}-1}{9 x \left (x^{2}-1\right )^{2}}\right ) u^{\prime }\left (x \right )+\frac {u \left (x \right )}{9 \left (x^{2}-1\right )^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \frac {2 \left (c_{2} x^{\frac {2}{3}} \pi \sqrt {3}\, \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+\frac {9 c_{1} \left (x^{2}\right )^{\frac {1}{3}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right )^{2}}{2}\right ) \sqrt {x^{2}-1}}{9 \left (x^{2}\right )^{\frac {1}{6}} \left (-x^{2}+1\right )^{\frac {5}{6}} \Gamma \left (\frac {2}{3}\right )} \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {\frac {2 \sqrt {3}\, \pi \left (x^{\frac {5}{3}}+x^{\frac {11}{3}} \left (x^{4}-x^{2}-1\right )\right ) c_{2} \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )}{9}+\left (x -1\right )^{2} c_{1} x \Gamma \left (\frac {2}{3}\right )^{2} \left (x^{2}\right )^{\frac {1}{3}} \left (x^{2}+1\right ) \left (x +1\right )^{2} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+\frac {14 \sqrt {3}\, \pi \left (-x^{\frac {5}{3}}+x^{\frac {11}{3}} \left (x^{4}-3 x^{2}+3\right )\right ) c_{2} \operatorname {LegendreP}\left (\frac {5}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )}{27}+\left (x -1\right )^{3} c_{1} x \Gamma \left (\frac {2}{3}\right )^{2} \left (x^{2}\right )^{\frac {1}{3}} \left (x +1\right )^{3} \operatorname {LegendreP}\left (\frac {5}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )}{2 \left (x^{2}\right )^{\frac {7}{6}} \left (-x^{2}+1\right )^{\frac {11}{6}} \left (x^{2}-1\right )^{\frac {3}{2}} \Gamma \left (\frac {2}{3}\right )} \] Using the above in (1) gives the solution \[ y = -\frac {27 \left (\frac {2 \sqrt {3}\, \pi \left (x^{\frac {5}{3}}+x^{\frac {11}{3}} \left (x^{4}-x^{2}-1\right )\right ) c_{2} \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )}{9}+\left (x -1\right )^{2} c_{1} x \Gamma \left (\frac {2}{3}\right )^{2} \left (x^{2}\right )^{\frac {1}{3}} \left (x^{2}+1\right ) \left (x +1\right )^{2} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+\frac {14 \sqrt {3}\, \pi \left (-x^{\frac {5}{3}}+x^{\frac {11}{3}} \left (x^{4}-3 x^{2}+3\right )\right ) c_{2} \operatorname {LegendreP}\left (\frac {5}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )}{27}+\left (x -1\right )^{3} c_{1} x \Gamma \left (\frac {2}{3}\right )^{2} \left (x^{2}\right )^{\frac {1}{3}} \left (x +1\right )^{3} \operatorname {LegendreP}\left (\frac {5}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )\right )}{4 x^{2} \left (-x^{2}+1\right ) \left (x^{2}-1\right ) \left (c_{2} x^{\frac {2}{3}} \pi \sqrt {3}\, \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+\frac {9 c_{1} \left (x^{2}\right )^{\frac {1}{3}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right )^{2}}{2}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {6 \pi \sqrt {3}\, \left (x^{\frac {2}{3}}+x^{\frac {8}{3}}\right ) \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+27 c_{3} \left (x^{2}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right )^{2} \left (x^{2}+1\right ) \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )-14 \pi \sqrt {3}\, \left (x^{\frac {2}{3}}-x^{\frac {8}{3}}\right ) \operatorname {LegendreP}\left (\frac {5}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+27 \operatorname {LegendreP}\left (\frac {5}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right )^{2} c_{3} \left (x^{2}\right )^{\frac {1}{3}} \left (x -1\right ) \left (x +1\right )}{18 x \left (c_{3} \left (x^{2}\right )^{\frac {1}{3}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right )^{2}+\frac {2 x^{\frac {2}{3}} \pi \sqrt {3}\, \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )}{9}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {6 \pi \sqrt {3}\, \left (x^{\frac {2}{3}}+x^{\frac {8}{3}}\right ) \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+27 c_{3} \left (x^{2}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right )^{2} \left (x^{2}+1\right ) \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )-14 \pi \sqrt {3}\, \left (x^{\frac {2}{3}}-x^{\frac {8}{3}}\right ) \operatorname {LegendreP}\left (\frac {5}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+27 \operatorname {LegendreP}\left (\frac {5}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right )^{2} c_{3} \left (x^{2}\right )^{\frac {1}{3}} \left (x -1\right ) \left (x +1\right )}{18 x \left (c_{3} \left (x^{2}\right )^{\frac {1}{3}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right )^{2}+\frac {2 x^{\frac {2}{3}} \pi \sqrt {3}\, \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )}{9}\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {6 \pi \sqrt {3}\, \left (x^{\frac {2}{3}}+x^{\frac {8}{3}}\right ) \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+27 c_{3} \left (x^{2}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right )^{2} \left (x^{2}+1\right ) \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )-14 \pi \sqrt {3}\, \left (x^{\frac {2}{3}}-x^{\frac {8}{3}}\right ) \operatorname {LegendreP}\left (\frac {5}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+27 \operatorname {LegendreP}\left (\frac {5}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right )^{2} c_{3} \left (x^{2}\right )^{\frac {1}{3}} \left (x -1\right ) \left (x +1\right )}{18 x \left (c_{3} \left (x^{2}\right )^{\frac {1}{3}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right )^{2}+\frac {2 x^{\frac {2}{3}} \pi \sqrt {3}\, \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )}{9}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 3 x \left (x^{2}-1\right ) y^{\prime }+x y^{2}-\left (x^{2}+1\right ) y=3 x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-x y^{2}+\left (x^{2}+1\right ) y+3 x}{3 x \left (x^{2}-1\right )} \end {array} \]
Maple trace Kovacic algorithm successful
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying Riccati trying Riccati sub-methods: trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -(1/3)*(5*x^2-1)*(diff(y(x), x))/(x*(x^2-1))+(1/3)*y(x)/(x^2-1)^2, y(x Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Tetrahedral Galois group A4_SL2. <- Kovacics algorithm successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 190
dsolve(3*x*(x^2-1)*diff(y(x),x) + x*y(x)^2 - (x^2+1)*y(x) - 3*x=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {80 c_{1} \sqrt {3}\, \pi \left (x^{2}-\frac {2}{5}\right ) \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+315 \Gamma \left (\frac {2}{3}\right ) \left (\frac {24 \left (x^{2}\right )^{\frac {1}{3}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right ) x^{\frac {4}{3}}}{35}+\left (x^{2}\right )^{\frac {1}{6}} \left (-x^{2}+1\right )^{\frac {5}{6}} \left (\left (x^{4}-x^{2}\right ) c_{1} \operatorname {hypergeom}\left (\left [\frac {11}{6}, \frac {13}{6}\right ], \left [\frac {7}{3}\right ], x^{2}\right )-\frac {6 \operatorname {hypergeom}\left (\left [\frac {3}{2}, \frac {11}{6}\right ], \left [\frac {5}{3}\right ], x^{2}\right ) \left (x^{\frac {4}{3}}-x^{\frac {10}{3}}\right )}{7}\right )\right )}{x^{\frac {1}{3}} \left (16 x^{\frac {2}{3}} \pi \sqrt {3}\, \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) c_{1} +72 \left (x^{2}\right )^{\frac {1}{3}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right )^{2}\right )} \]
✓ Solution by Mathematica
Time used: 4.513 (sec). Leaf size: 3149
DSolve[3*x*(x^2-1)*y'[x] + x*y[x]^2 - (x^2+1)*y[x] - 3*x==0,y[x],x,IncludeSingularSolutions -> True]
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